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"Young man, in mathematics you don't understand things. You just get used to them."
by vinchuco 10y ago
"Young man, in mathematics you don't understand things. You just get used to them."
- johnfn 10y agoI don't really like this, because it's a self limiting quote that leads to complacency over real understanding. Especially in the case of imaginary numbers and e where intuitive understanding only doesn't exist because mathematics has a history of being poorly taught.
- E6300 10y agoHonestly, after spending months studying the subject, I don't think it's really possible to "get" complex numbers. I just view them as affine transformations written in an unusual notation. I don't think they make sense as anything but a recontextualization of R^2.
- akvadrako 10y agoSo what don't you get? Are you implying there is something more to complex numbers?
- E6300 10y agoWhat I don't get is why someone would bother with complex numbers and their silly notation when linear algebra would work perfectly fine. I know that in certain contexts they are useful. E.g. to avoid losing information when solving polynomials. But that's a minuscule fraction of their range of application. Nearly every practical use I've seen of complex numbers just use them as a vector representation.
- akvadrako 10y agoWell the two uses I'm most familiar with are AC circuit analysis and quantum mechanics. They can both be reformulated without complex numbers of course, since nothing is special about i. Yet the complex versions are a lot easier to work with, because even in manifestly real formulations, the complex structure is still there, but in disguise: - http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html - http://physics.stackexchange.com/questions/32422/qm-without-complex-numbers http://physics.stackexchange.com/questions/32422/qm-without-...
- jameshart 10y agoThe problem with that is that complex numbers initially emerge as roots of polynomials with real coefficients. Getting to affine transformations from there seems a much bigger leap than asserting there is a square root of -1.
- E6300 10y agoThe video linked in this post will only make sense if you accept that x e^(theta i) and x k are respectively the rotation part and the scaling part of a linear transformation of x. I'm not aware of any other way to intuitively grasp an expression like e^(pi i).
- ouid 10y agothis is an old comment, but e^z (over C) can be defined as the analytic continuation of e^x (over R). This is a much more interesting definition, since it depends on the fact that that function is unique.
- snogglethorpe 10y agoMild snarkiness aside, is the quote really inaccurate? "Getting used to" [something] has a pejorative sense, but it also just means "becoming familiar with," and really understanding something is, in a way, simply being so familiar with it that reasoning about it is second nature... at some point, things just sort of start to make sense...
- hueving 10y agoNot quite, you can get used to something without understanding it at all. Many people that fly are used to it but they don't understand the basic physics of flight at all.
- AndrewOMartin 10y agoAnd who would you prefer in the pilot's seat? To say suggest that you don't understand something without a formal underlying theory is one view, but not mine. I feel like I understand English (as in a deep understanding, not just the ability to interpret sentences) without anything like a set of rules, and everything like a set of experiences akin to a pilot's experience with flight.
- hueving 10y agoWhat does that have to do with anything I said? I'm pointing out the difference between "understanding" and "getting used to".
- AndrewOMartin 10y agoI'm refuting that difference.
- hueving 10y agoNot very well. I'm not talking about pilots. I fly with many frequent fliers that are completely comfortable with flying while having no technical understanding of the mechanics.
- pyromine 10y agoI think there is both a good and bad aspect of this quote, you've done a great job highlighting the bad. The good, is in essence that you don't always need to be fully comfortable and in fact will not always be comfortable. I find in mathematics if you try to always strive for total comfort you will never progress, as a lot of the comfort comes from advancing past a topic and upon revisiting it you realize you understand the fundamentals better than you thought.
- bsaul 10y agoTotally agree. My father would always tell that what's important is to truely understand things and not just be able to do the problem. This had two bad side effects : 1/ i just read the course and didn't do the exercises, and 2/ i "blocked" on things like infinitesimal calculus, because i couldn't get my head on its true meaning.
- espeed 10y agoRe: knowing vs understanding, have you read "Richard Feynman on education in Brazil"? http://v.cx/2010/04/feynman-brazil-education http://v.cx/2010/04/feynman-brazil-education
- johnfn 10y agoAgreed, but isn't that irrelevant here? e^i*pi has an intuitive meaning, so we shouldn't be content with our lack of understanding.
- askvictor 10y agoIn every maths subject at uni, I'd do quite well in the exams, but not fully 'getting it'. Then the next semester, in the next subject along the, I suddenly understood it, as if the semester break had given things a chance to arrange themselves, and the context of the more advanced subject made things clear.